Preliminary Exam - Fall 1993

Problem 1   Let $X$ be a metric space and $(x_{n})$ a convergent sequence in $X$ with limit $x_{0}$. Prove that the set $C = \{x_0, x_1, x_2,...\}$ is compact.

Problem 2   Let $A$ be the additive group of rational numbers, and let $M$ be the multiplicative group of positive rational numbers. Determine all homomorphisms of $A$ into $M$.

Problem 3   Describe the region in the complex plane where the infinite series

$$% latex2html id marker 690 \sum^{\infty}_{n=1} \frac{1}{n^{2}} \exp \!\left(\frac{nz}{z-2}\right)$$

converges. Draw a sketch of the region.

Problem 4   Let $\mbox{\bf {F}}$ be a field. For $m$ and $n$ positive integers, let $M_{m\times n}$ be the vector space of $m \times n$ matrices over $\mbox{\bf {F}}$. Fix $m$ and $n$, and fix matrices $A$ and $B$ in $M_{m\times n}$. Define the linear transformation $T$ from $M_{n\times m}$ to $M_{m\times n}$ by

$$T(X) = AXB.$$

Prove that if $m \neq n$, then $T$ is not invertible.

Problem 5   Let the function $x(t) \; (-\infty < t < \infty)$ be a solution of the differential equation

$$\frac{d^{2}x}{dt^{2}} - 2b \, \frac{dx}{dt} + cx = 0$$

such that $x(0) = x(1) = 0$. (Here, $b$ and $c$ are real constants.) Prove that
$x(n) =0$ for every integer $n$.

Problem 6   Let $A$, $B$, and $C$ be finite abelian groups. Prove that if $A \times B$ is isomorphic to $A \times C$, then $B$ is isomorphic to $C$.

Problem 7   Let $M_{n \times n}$ $(n \geq 2)$ be the space of real n$\times$n matrices, identified in the usual way with the Euclidean space $\mathbb{R}^{n^{2}}$. Let $F$ be the determinant map of $M_{n \times n}$ into $\mathbb{R}$: $F(X)=\det (X)$. Find all of the critical points of $F$; that is, all matrices $X$ such that $DF(X) = 0$. _function,>critical points

Problem 8   Prove that if $A$ is an $n \times n$ matrix over $\mathbb{C}$, and if $A^k = I$ for some positive integer $k$, then $A$ is diagonalizable.

Problem 9   Evaluate

$$\int^{\infty}_{-\infty} \ \frac{e^{-ix}}{x^{2} - 2x + 4} \ dx\, .$$

Problem 10   Let $f$ be a continuous real valued function on $[0, \infty)$. Let $A$ be the set of real numbers $a$ that can be expressed as $a = \lim_{n \to \infty} f(x_{n})$ for some sequence $(x_{n})$ in $[0, \infty)$ such that $\lim_{n \to \infty} x_n = \infty$. Prove that if $A$ contains the two numbers $a$ and $b$, then it contains the entire interval with endpoints $a$ and $b$.

Problem 11   Let $R$ be a commutative ring with identity. Let $G$ be a finite subgroup of $R^*$, the group of units of $R$. Prove that if $R$ is an integral domain, then $G$ is cyclic.

Problem 12   Evaluate $\frac{1}{2\pi i} \int_{\gamma} f(z) \ dz$ for the function
$f(z) = z^{-2}(1 - z^{2})^{-1}e^z$ and the curve $\gamma$ depicted by

file=../Fig/Pr/Fa93-12,width=3.7in

Problem 13   Show that there are at least two nonisomorphic nonabelian groups of order $40$.

Problem 14   Let $n$ be an integer larger than $1$. Is there a differentiable function on $[0,\infty)$ whose derivative equals its $n^{th}$ power and whose value at the origin is positive?

Problem 15   Prove that the matrix

$$\left( \begin{array}{cccc} 0 & 5 & 1 & 0 \\ 5 & 0 & 5 & 1 \\ 1 & 5 & 0 & 5 \\ 0 & 1 & 5 & 0 \end{array} \right)$$

has two positive and two negative eigenvalues (taking into account multiplicities).

Problem 16   Let $K$ be a continuous real valued function on $[0,1] \times [0,1]$. Let $F$ be the family of functions $f$ on $[0,1]$ of the form

$$f(x) = \int^1_0 \ g(y)K(x,y) \, dy$$

with $g$ a real valued continuous function on $[0,1]$ satisfying $\vert g\vert \leq 1$ everywhere. Prove that the family $F$ is equicontinuous.

Problem 17   Let $w$ be a positive continuous function on $[0,1]$, $n$ a positive integer, and $P_n$ the vector space of real polynomials whose degrees are at most $n$, equipped with the inner product

$$\langle p,q \rangle = \int^1_{0} \ p(t)q(t)w(t) \ dt \, .$$

1. Prove that $P_n$ has an orthonormal basis $p_0,$ $p_1,$ $\ldots,$ $p_n$ (i.e., $\langle p_j, p_k \rangle = 1$ for $j = k$ and $0$ for $j \neq k$) such that $\deg p_k = k$ for each $k$.

2. Prove that $\langle p_k, p^\prime _k \rangle = 0$ for each $k$.

Problem 18   Let $f$ be a continuous real valued function on $[0,1]$, and let the function $h$ in the complex plane be defined by

$$h(z) = \int^1_{0} \ f(t) \cos zt \, dt.$$

1. Prove that $h$ is analytic in the entire plane.
2. Prove that $h$ is the zero function only if $f$ is the zero function.